惯性尺度下分数耗散的张量扩展

中文摘要 AI 辅助

我们提出了一个由宏观纳维-斯托克斯对应关系启发的分数耗散算子的尺度依赖张量扩展。起点是一个分数模型，其中耗散阶数随着局部雷诺数的增加而从拉普拉斯值连续降低到柯尔莫哥洛夫值。在局部雷诺数的临界值处，该模型简化为各向同性算子。然后我们询问，是否可以在不引入无关闭合或额外各向同性耗散尺度的情况下，将相同的宏观描述延续到速度场具有Onsager、临界或次临界正则性的区域。此外，我们通过一个非负浓度序参数c扩展了原始的AFNS自由能构造。新项在c中是二次的，并与缺陷β函数线性耦合，因此当浓度分支不活跃时，最小化使原始AFNS自由能完全不变。由此产生的三区域马赛克是正则的、惯性的和浓缩/异常的。该构造使用粗粒化应力及其临界归一化。归一化各向异性张量提供了临界尺度上可用的最低阶客观张量信息，而Duchon Robert能量缺陷的尺度导数提供了偏离Onsager阈值的标量度量。我们提出了临界算子的最小本构变形：在Weyl量化中，各向同性分数符号被这个指数张量因子修饰，得到一个伪微分算子，它简化为标量临界算子。指数参数化保证了对称无迹算子的正特征值和单位行列式，因此该变形在方向之间重新分配耗散，而不引入新的各向同性幅度。

英文摘要

We formulate a scale dependent tensorial extension of a fractional dissipation operator motivated by a macroscopic Navier Stokes correspondence. The starting point is a fractional model in which the order of dissipation decreases continuously from the Laplacian value toward the Kolmogorov value as a local Reynolds number increases. At the critical value of the local Reynolds number, the model reduces to an isotropic operator. We then ask whether the same macroscopic description can be continued into a regime in which velocity fields have Onsager, critical or subcritical regularity without introducing an unrelated closure or an additional isotropic dissipation scale. In addition, we extend the original AFNS free, energy, construction by one nonnegative concentration order parameter c. The new term is quadratic in c and linearly coupled to the defect beta function, so that minimization leaves the original AFNS free energy exactly unchanged when the concentration branch is inactive. The resulting three regime mosaic is regular, inertial, and concentrated/anomalous. The construction uses the coarse-grained stress and its critical normalization. The normalized anisotropy tensor provides the lowest order objective tensorial information available at the critical scale, while the scale derivative of the Duchon Robert energy defect, supplies a scalar measure of departure from the Onsager threshold. We propose the minimal constitutive deformation of the critical operator: in Weyl quantization, the isotropic fractional symbol is dressed by this exponential tensorial factor, giving a pseudodifferential operator that reduces to the scalar critical operator. The exponential parametrization guarantees positive eigenvalues and unit determinant for symmetric traceless, so the deformation redistributes dissipation among directions without introducing a new isotropic amplitude.

发表机构

• University of São Paulo（圣保罗大学）

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